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A statistical mechanical interpretation of
algorithmic information theory
Kohtaro Tadaki
Research and Development Initiative, Chuo University
CREST, JST
Tokyo, Japan
Supported by SCOPE from the Ministry of Internal Affairs and Communications of Japan
1
Abstract
In this talk, we develop a statistical mechanical interpretation of Algorithmic
Information Theory (AIT, for short).
We introduce the notion of thermodynamic quantities, such as free energy, energy, (statistical mechanical) entropy, and specific heat, into AIT.
We then investigate their properties from the point of view of algorithmic
randomness. As a result, we see that, in this statistical mechanical interpretation, the temperature equals to the partial randomness (and therefore
compression rate) of the values of all these thermodynamic quantities, which
include the temperature itself.
Reflecting this self-referential nature of the temperature, we obtain a fixed
point theorems on partial randomness.
2
Preliminaries: Program-size Complexity
• {0, 1}∗ := {λ, 0, 1, 00, 01, 10, 11, 000, . . . }. The set of finite binary strings.
• For any s ∈ {0, 1}∗, |s| denotes the length of s.
• Let V ⊂ {0, 1}∗. We say V is a prefix-free set if for any distinct s and
t ∈ V , s is not a prefix of t.
For example
{0, 10}: prefix-free
{0, 01}: not prefix-free
U : universal self-delimiting Turing machine.
Dom U , i.e., the domain of definition of U , is a prefix-free set.
Definition
The program-size complexity (or Kolmogorov complexity) H(s)
of s ∈ {0, 1}∗ is defined by
H(s) := min
{
¯
}
¯
∗
|p| ¯ p ∈ {0, 1} & U (p) = s .
H(s): The length of the shortest input for the universal self-delimiting Turing machine U to output s.
⇒ H(s): The degree of randomness of s.
(the size of compressed s)
3
Preliminaries: Randomness of Real Number
Let α ∈ R. α ¹ n denotes the first n bits of the base-two expansion of α−bαc.
The fractional part of α.
Definition [weak Chaitin randomness, Chaitin 1975]
We say α ∈ R is weakly Chaitin random if n ≤ H(α ¹ n) + O(1),
i.e., any prefix of the base-two expansion of α cannot be compressed by H.
This notion is equivalent to Martin-Löf randomness.
Definition [Chaitin’s halting probability Ω, Chaitin 1975]
Ω :=
∑
2−|p|.
p∈Dom U
The first n bits of the base-two expansion of Ω solve the halting problem
of U for inputs of length at most n.
Theorem [Chaitin 1975] Ω is weakly Chaitin random.
4
Preliminaries: Partial Randomness of Real Number
The partial randomness (degree of randomness) of a real can be characterized by a real.
Definition [weak Chaitin D-randomness, Tadaki 2002] Let D ∈ [0, 1].
We say α ∈ R is weakly Chaitin D-random if Dn ≤ H(α ¹ n) + O(1) for ∀ n.
In the case of D = 1, the weak Chaitin D-randomness results in the weak
Chaitin randomness.
Definition [D-compressibility] Let D ∈ [0, 1].
We say α ∈ R is D-compressible if H(α ¹ n) ≤ Dn + o(n),
Remark
H(α ¹ n)
which is equivalent to lim sup
≤ D.
n→∞
n
If α ∈ R is weakly Chaitin D-random and D-compressible, then
H(α ¹ n)
= D.
n→∞
n
The compression rate of α by program-size complexity is equal to D.
hThe converse does not necessarily hold.i
lim
5
Preliminaries: Generalization of Ω
Definition [generalization of Chaitin’s Ω, Tadaki 1999]
Ω(D) :=
∑
|p|
−D
2
(D > 0).
p∈Dom U
Ω(1) = Ω. If 0 < D ≤ 1, then Ω(D) ≤ Ω and Ω(D) converges, in particular.
Theorem [Tadaki 1999] Let D ∈ R.
(i) If 0 < D ≤ 1 and D is computable, then Ω(D) is weakly Chaitin Drandom and D-compressible.
⇒ The partial randomness (compression rate) of Ω(D) equals to D.
(ii) If 1 < D, then Ω(D) diverges to ∞.
Here, we say that a real D is computable if the mapping N+ 3 n 7→ D ¹ n is
a total recursive function.
6
Motivation
[Calude & Stay, Information and Computation 204 (2006)] pointed out that
Ω(D) is similar to a partition function in statistical mechanics.
• In statistical mechanics, the partition function Z is given as:
Z=
∑ − En
kT
e
.
n
Here, n denotes the quantum number of an energy eigenstate of a quantum
system, En its energy, and T temperature.
• On the other hand, Ω(D) is given as:
∑
Ω(D) =
|p|
−D
2
(D > 0).
p∈Dom U
Thus, Z coincides with Ω(D) by performing the following replacements:
An energy eigenstate n
The energy En of n
Temperature T
Boltzmann constant k
⇒
⇒
⇒
⇒
A program p ∈ Dom U ,
The length |p| of p,
Compression rate D,
1/ ln 2.
7
What is the partition function in statistical mechanics ?
Quick Review of Statistical Mechanics (I)
Consider a quantum system at constant temperature T .
(That is, imagine a quantum system in thermal contact with a very large
heat reservoir at constant temperature T )
Statistical mechanics states that the probability Prob(n) that the system is
in an energy eigenstate n with energy En is given as:
1 − En
Prob(n) = e kT .
Z
Here, the normalization factor Z :=
∑ − En
kT
e
is called the partition function
n
of the quantum system.
The partition function Z is of particular importance in statistical mechanics, because all the thermodynamic quantities of the quantum system can
be expressed by using the partition function Z, and the knowledge of Z is
sufficient to understand all the macroscopic properties of the system.
8
Quick Review of Statistical Mechanics (II)
Thermodynamic Quantities of the quantum system at temperature T
• Free Energy
F = −kT ln Z.
The free energy F of a quantum system is related to the work performed
by the system during a process at constant temperature T .
• Energy
∑
1∑
d
n
−E
2
kT
E=
En Prob(n) =
En e
= kT
ln Z.
Z n
dT
n
Thus, the energy E of the quantum system is the expected value of an
energy En of an energy eigenstate n.
• (Statistical Mechanical) Entropy
E−F
S=
.
T
Note that the entropy S of the system equals to the
∑ Shannon entropy of
the probability distribution {Prob(n)}, i.e., S = −k
Prob(n) ln Prob(n).
n
• Specific Heat
dE
.
C=
dT
9
Aim of this talk
We propose a statistical mechanical interpretation of AIT (Algorithmic Information Theory) where Ω(D) appears as a partition function.
We do this in the following manner:
We introduce the notion of thermodynamic quantities such as free energy,
energy, (statistical mechanical) entropy, and specific heat into AIT by performing the following replacements for the corresponding thermodynamic
quantities of a quantum system at temperature T :
An energy eigenstate n
The energy En of n
Boltzmann constant k
⇒
⇒
⇒
A program p ∈ Dom U ,
The length |p| of p,
1/ ln 2.
We then determine the convergence or divergence of each of the quantities.
In the case where a thermodynamic quantity converges, we calculate the
partial randomness (compression rate) of the values of the thermodynamic
quantity, based on program-size complexity H(s).
⇒ We see that all of the partial randomness of the thermodynamic quantities, which include temperature T itself, equal to T .
10
Immediate Application of the Replacements: Transient Definitions
Perform the following replacements for the corresponding thermodynamic
quantities of a quantum system at temperature T . (U : optimal computer)
The energy En of n
⇒
⇒
Boltzmann constant k
⇒
An energy eigenstate n
Partition function
Z(T ) =
A program p ∈ Dom U ,
The length |p| of p,
1/ ln 2.
∑ − En
kT
⇒
e
n
Free energy
Energy
Entropy
S(T ) =
Specific heat
E(T ) − F (T )
T
C(T ) =
d
E(T )
dT
Z(T ) =
∑
2
|p|
−T
,
p∈Dom U
⇒
F (T ) = −kT ln Z(T )
1 ∑
n
−E
Ene kT
E(T ) =
Z(T ) n
Boltzmann factor: 2
⇒
⇒
⇒
F (T ) = −T log2 Z(T ),
|p|
∑
1
−T
|p| 2
,
E(T ) =
Z(T ) p∈Dom U
S(T ) =
E(T ) − F (T )
,
T
C(T ) =
d
E(T ).
dT
11
|p|
−T
Thermodynamic Quantities in AIT: Rigorous Definitions
Redefine the transient definitions rigorously as follows.
Definition
Let q1, q2, q3, . . . . . . be an arbitrary enumeration of Dom U .
Note that the results of this talk are independent of the choice of {qi}.
Definition [Thermodynamic Quantities in AIT, Tadaki 2008] Let T > 0.
(i) partition function Z(T ) := lim Zm(T ), where Zm(T ) =
m→∞
m
∑
|q |
− Ti
2
.
i=1
(ii) free energy F (T ) := lim Fm(T ), where Fm(T ) = −T log2 Zm(T ).
m→∞
m
|q |
∑
1
− Ti
|qi| 2
(ii) energy E(T ) := lim Em(T ), where Em(T ) =
.
m→∞
Zm(T ) i=1
(iii) entropy S(T ) := lim Sm(T ), where Sm(T ) =
m→∞
Em(T ) − Fm(T )
.
T
0 (T ).
(iv) specific heat C(T ) := lim Cm(T ), where Cm(T ) = Em
m→∞
Remark
These are variants of Chaitin’s Ω. In particular, Z(T ) = Ω(T ).
12
Temperature = Partial Randomness.
13
Statistical Mechanical Quantity in AIT: Partition Function
Recall that, in statistical mechanics, the partition function Z is given by:
Z=
∑ − En
kT
e
.
n
Definition [partition function] For each m ∈ N+ and each real T > 0,
Zm(T ) of finite size m is defined as
Zm(T ) :=
m
∑
|q |
− Ti
2
.
i=1
Then, for each T > 0, the partition function Z(T ) is defined as
Z(T ) := lim Zm(T ).
m→∞
Z(T ) = Ω(T ).
Theorem [partition function, Tadaki 1999, 2002] Let T ∈ R.
(i) If 0 < T ≤ 1 and T is computable, then Z(T ) converges to a leftcomputable real which is weakly Chaitin T -random and T -compressible.
⇒ The partial randomness of Z(T ) equals to the temperature T .
(ii) If 1 < T , then Z(T ) diverges to ∞.
⇒ “Phase Transition” occurs at temperature 1.
14
Thermodynamic Quantities in AIT: Free Energy
Recall that, in statistical mechanics, the free energy F is given by:
F = −kT ln Z.
Definition [free energy] For each m ∈ N+ and each real T > 0, Fm(T ) of
finite size m is defined as
Fm(T ) := −T log2 Zm(T ).
Then, for each T > 0, the free energy F (T ) is defined as
F (T ) := lim Fm(T ).
m→∞
Theorem [free energy] Let T ∈ R.
(i) If 0 < T ≤ 1 and T is computable, then F (T ) converges to a rightcomputable real which is weakly Chaitin T -random and T -compressible.
⇒ The partial randomness of F (T ) equals to the temperature T .
(ii) If 1 < T , then F (T ) diverges to −∞.
⇒ “Phase Transition” occurs at temperature 1.
15
Thermodynamic Quantities in AIT: Energy
Recall that, in statistical mechanics, the energy E is given by:
1∑
n
−E
kT
E=
Ene
.
Z n
Definition [energy] For each m ∈ N+ and each real T > 0, Em(T ) of finite
size m is defined as
m
|q |
∑
1
− Ti
|qi| 2
Em(T ) :=
=
Zm(T ) i=1
∑m
−|qi |/T
i=1 |qi | 2
.
∑m
−|q
|/T
i
i=1 2
Then, for each T > 0, the energy E(T ) is defined by E(T ) := limm→∞ Em(T ).
Definition
We say α ∈ R is Chaitin T -random if limn→∞ H(α ¹ n)−T n = ∞.
Theorem [energy] Let T ∈ R.
(i) If 0 < T < 1 and T is computable, then E(T ) converges to a leftcomputable real which is Chaitin T -random and T -compressible.
⇒ The partial randomness of E(T ) equals to the temperature T .
(ii) If 1 ≤ T , then E(T ) diverges to ∞.
⇒ “Phase Transition” occurs at temperature 1.
16
Thermodynamic Quantities in AIT: Entropy
Recall that, in statistical mechanics, the entropy S is given by:
E−F
S=
.
T
Definition [entropy] For each m ∈ N+ and each real T > 0, Sm(T ) of finite
size m is defined as
Em(T ) − Fm(T )
.
T
Then, for each T > 0, the entropy S(T ) is defined as
Sm(T ) :=
S(T ) := lim Sm(T ).
m→∞
Theorem [entropy] Let T ∈ R.
(i) If 0 < T < 1 and T is computable, then S(T ) converges to a leftcomputable real which is Chaitin T -random and T -compressible.
⇒ The partial randomness of S(T ) equals to the temperature T .
(ii) If 1 ≤ T , then S(T ) diverges to ∞.
⇒ “Phase Transition” occurs at temperature 1.
17
Thermodynamic Quantities in AIT: Specific Heat
Recall that, in statistical mechanics, the specific heat C is given by:
dE
C=
.
dT
Definition [specific heat] For each m ∈ N+ and each real T > 0, Cm(T )
of finite size m is defined as 

∑
2
∑


2 −|qi|/T
m |q | 2−|qi |/T

|q
|
2
d
ln 2  m
i
i
i=1
i=1
 ∑

Cm(T ) :=
Em(T ) = 2
−
.
∑
m
m
−|q
|/T
−|q
|/T


i
i
dT
T 

i=1 2
i=1 2
Then, for each T > 0, the specific heat C(T ) is defined as
C(T ) := lim Cm(T ).
m→∞
Theorem [specific heat] Let T ∈ R.
(i) If 0 < T < 1 and T is computable, then C(T ) converges to a leftcomputable real which is Chaitin T -random and T -compressible.
⇒ The partial randomness of C(T ) equals to the temperature T .
(ii) If T = 1, then C(T ) diverges to ∞.
⇒ “Phase Transition” occurs at temperature 1.
In the case of T > 1, it is still open whether C(T ) diverges or not.
18
Thermodynamic Quantities in AIT: Properties in Summary
Definition We say α ∈ R is weakly Chaitin T -random if H(α ¹ n) − T n ≥
O(1).
Theorem [partition function Z(T ), free energy F (T )] Let T ∈ R.
(i) If 0 < T ≤ 1 and T is computable, then each of Z(T ) and F (T ) converges
to real which is weakly Chaitin T -random and T -compressible.
(ii) If 1 < T , then Z(T ) and F (T ) diverge to ∞ and −∞, respectively.
Definition
We say α ∈ R is Chaitin T -random if limn→∞ H(α ¹ n)−T n = ∞.
Theorem [energy E(T ), entropy S(T ), specific heat C(T )] Let T ∈ R.
(i) If 0 < T < 1 and T is computable, then each of E(T ), S(T ), and C(T )
converges to real which is Chaitin T -random and T -compressible.
(ii) If 1 ≤ T , then both E(T ) and S(T ) diverge to ∞. In the case of T = 1,
C(T ) diverge to ∞.
Implication of (i): The partial randomness of the values of the thermodynamic quantities equals to the temperature T .
Thermodynamic Interpretation of (ii): “Phase Transition” occurs at temperature 1.
19
Thermodynamic Quantities in AIT: Temperature
Temperature ⇒ Fixed Point Theorems
In the case where T is computable with 0 < T < 1,
all of the partial randomness of the thermodynamic quantities:
partition function Z(T ), free energy F (T ),
energy E(T ), entropy S(T ),and specific heat C(T ),
equal to the temperature T .
However,
one of the most typical thermodynamic quantities is temperature T itself.
Thus, the following question arises naturally:
Question
Can the partial randomness of the temperature equal to
the temperature itself ?
Self-referential Question
We can answer this question affirmatively in the following form:
20
Fixed Point Theorem on Partial Randomness: Main Theorem
Theorem [fixed point theorem on partial randomness, Tadaki, CiE 2008]
For every T ∈ (0, 1), if Z(T ) is a computable real, then
(i) T is weakly Chaitin T -random and T -compressible, and therefore
⇒ The partial randomness of T equals to T itself.
H(T ¹ n)
(ii) lim
= T.
n→∞
n
⇒ The compression rate of T equals to T itself.
Intuitive Meaning; Metaphor
Consider a file of infinite size whose content is
“The compression rate of this file is 0.100111001 · · · · · · ”
When this file is compressed, the compression rate of this file actually equals
to 0.100111001 · · · · · · , as the content of this file says.
This situation forms a fixed point and is self-referential !
21
Fixed Point Theorem on Partial Randomness: Proof
Proof of Fixed Point Theorem
Theorem [fixed point theorem on partial randomness]
[posted again]
For every T ∈ (0, 1), if Z(T ) is a computable real, then
(i) T is right-computable and not left-computable,
(ii) T is weakly Chaitin T -random and T -compressible,
H(T ¹ n)
(iii) lim
=T
(and therefore T is not computable).
n→∞
n
Lemma [upper bound I ] For every T ∈ (0, 1), if Z(T ) is right-computable
then T is also right-computable.
Lemma [upper bound II ] For every T ∈ (0, 1), if Z(T ) is left-computable
and T is right-computable, then T is T -compressible.
Lemma [lower bound] For every T ∈ (0, 1), if Z(T ) is right-computable
then T is weakly Chaitin T -random.
22
Proofs of the three lemmas
Lemma [upper bound I ] For every T ∈ (0, 1), if Z(T ) is right-computable
then T is also right-computable.
∑k
+
For each k ∈ N and each x ∈ (0, 1), let ωk (x) = i=1 2−|pi|/x,
Proof)
where p1, p2, p3, . . . . . . is a particular recursive enumeration of Dom U .
Then we see that, for every r ∈ Q ∩ (0, 1), T < r if and only if there exists
∑∞
+
k ∈ N such that Z(T ) < ωk (r). This is because Z(x) = i=1 2−|pi|/x is an
increasing function of x ∈ (0, 1] and limk→∞ ωk (r) = Z(r).
Since Z(T ) is right-computable,
the set { r ∈ Q ∩ (0, 1) | ∃ k ∈ N+ Z(T ) < ωk (r) } is r.e. and therefore
the set { r ∈ Q ∩ (0, 1) | T < r } is also r.e.
Lemma [upper bound II ] For every T ∈ (0, 1), if Z(T ) is left-computable
and T is right-computable, then T is T -compressible.
Proof) Omitted.
23
Lemma [lower bound] For every T ∈ (0, 1), if Z(T ) is right-computable
then T is weakly Chaitin T -random.
Proof) The following procedure calculates a partial recursive function
Ψ : {0, 1}∗ → {0, 1}∗ such that T n − T c < H(Ψ(T ¹ n)). The lemma fol∑
lows from H(Ψ(T ¹ n)) ≤ H(T ¹ n) + O(1).
Let ωk (x) = ki=1 2−|pi|/x.
Procedure: Given T ¹ n, one can effectively find k0 which satisfies
Z(T ) < ωk0 (0.(T ¹ n) + 2−n).
This is possible because Z(x) is an increasing function of x, limk→∞ ωk (r) =
Z(r) for every r ∈ Q ∩ (0, 1), and Z(T ) is right-computable. It follows that
∞
∑
2
|p |
− Ti
= Z(T ) − ωk0 (T ) < ωk0 (0.(T ¹ n) + 2−n) − ωk0 (T ) < 2c−n.
i=k0+1
|p |
− Ti
Hence, for every i > k0, 2
< 2c−n and therefore T n − T c < |pi|. Thus,
by calculating the set { U (pi) | i ≤ k0 } and picking any one finite binary
string s which is not in this set, one can then obtain s ∈ {0, 1}∗ such that
T n − T c < H(s).
24
Remark on the sufficient condition in the fixed Point Theorem
Theorem [fixed point theorem on partial randomness]
[posted again]
For every T ∈ (0, 1), if Z(T ) is computable, then T is weakly Chaitin T random and T -compressible.
∑
−|qi |/x is a strictly increasing continuous function
Note that Z(x) = ∞
i=1 2
of x ∈ (0, 1), and the set of all computable reals is dense in R. Thus,
The set {T ∈ (0, 1) | Z(T ) is computable } is dense in (0, 1).
Theorem
Corollary [density of the fixed points]
The set {T ∈ (0, 1) | T is weakly Chaitin T -random and T -compressible} is
dense in (0, 1).
At this point, the following question would arise naturally:
Is this sufficient condition, i.e., the computability of Z(T ),
Question
also necessary for T to be a fixed point ?
Answer
Completely not !!
(as we can see through the following
argument)
25
Thermodynamic Quantities in AIT: Fixed Point Theorems
In the fixed point theorem, Z(T ) can be replaced by each of the thermodynamic quantities F (T ), E(T ), and S(T ) as follows.
Theorem [fixed point theorem by the free energy F (T ), Tadaki 2009]
For every T ∈ (0, 1), if F (T ) is computable, then
(i) T is weakly Chaitin T -random and T -compressible, and therefore
(ii) limn→∞ H(T ¹ n)/n = T .
This fixed point theorem has the exactly same form as by Z(T ).
Theorem [fixed point theorem by the energy E(T ), Tadaki 2009]
For every T ∈ (0, 1), if E(T ) is computable, then
(i) T is Chaitin T -random and T -compressible, and therefore
(ii) limn→∞ H(T ¹ n)/n = T .
Theorem [fixed point theorem by the entropy S(T ), Tadaki 2009]
For every T ∈ (0, 1), if S(T ) is computable, then
(i) T is Chaitin T -random and T -compressible, and therefore
(ii) limn→∞ H(T ¹ n)/n = T .
26
Proofs of the fixed point theorems by F (T ), E(T ), and S(T )
Lemma [thermodynamic relations] T ∈ (0, 1).
(i) Fk0 (T ) = −Sk (T ), Ek0 (T ) = Ck (T ), and Sk0 (T ) = Ck (T )/T .
(ii) F 0(T ) = −S(T ), E 0(T ) = C(T ), and S 0(T ) = C(T )/T .
(iii) Sk (T ), Ck (T ) ≥ 0. Sk (T ), Ck (T ) > 0 for all sufficiently large k.
S(T ), C(T ) > 0.
A portion of the proof of the above lemma :
In particular, the condition (iii) is proved based on the statistical mechanical
relations:
There exists k0 ∈ N+ such that, for every k ≥ k0 and every T ∈ (0, 1),
|pi |
|pi |
2− T
log2
> 0,
Sk (T ) = −
Zk (T )
i=1 Zk (T )
k
∑
2−
T
Shannon entropy of
|pi |
−
k
ln 2 ∑
2 T
2
> 0.
Ck (T ) = 2
{|pi| − Ek (T )}
T i=1
Zk (T )

|p |


 2− Ti








 Zk (T ) 
Variance of energy Ek (T )
27
i
Relation between the sufficient conditions of FPTs
Theorem
There does not exist T ∈ (0, 1) such that both Z(T ) and F (T )
are computable.
Proof)
Contrarily, assume that both Z(T ) and F (T ) are computable for some T ∈
(0, 1). Since the statistical mechanical relation F (T ) = −T log2 Z(T ) holds,
T =−
F (T )
.
log2 Z(T )
Thus, T is computable, and therefore Z(T ) is weakly Chaitin T -random,
i.e., T n ≤ H(Z(T ) ¹ n) + O(1). However, this is impossible, since Z(T ) is
computable by the assumption, and therefore H(Z(T ) ¹ n) ≤ 2 log2 n+O(1).
Thus we have a contradiction.
{T ∈ (0, 1) | Z(T ) is computable } ∩ {T ∈ (0, 1) | F (T ) is computable } = ∅.
dense in (0, 1)
dense in (0, 1)
In particular, this shows that the computability of Z(T ) is not a necessary
condition for T to be a fixed point in the fixed point theorem by Z(T ).
28
Summary
Partial Randomness = Temperature.
(Compression Rate)
29
Remark: Mathematical Implication of the Results
The proofs of the fixed point theorems on partial randomness depend heavily on the following thermodynamic relations:
Lemma [thermodynamic relations] T ∈ (0, 1).
(i) Fk0 (T ) = −Sk (T ), Ek0 (T ) = Ck (T ), and Sk0 (T ) = Ck (T )/T .
(ii) F 0(T ) = −S(T ), E 0(T ) = C(T ), and S 0(T ) = C(T )/T .
(iii) Sk (T ), Ck (T ) ≥ 0. Sk (T ), Ck (T ) > 0 for all sufficiently large k.
S(T ), C(T ) > 0.
Moreover, the proof of the following theorem depends on the statistical
mechanical relation F (T ) = −T log2 Z(T ).
Theorem
There does not exist T ∈ (0, 1) such that both Z(T ) and F (T )
are computable.
This theorem says that the computability of F (T ) gives completely different
fixed points from the computability of Z(T ).
These fact would imply that the analytic method can be used in the research
of AIT (algorithmic randomness).
30
Remark: Physical Implication of the Results
Definition
Let q1, q2, q3, . . . . . . be an arbitrary enumeration of Dom U .
In the statistical mechanical interpretation of AIT,
q1, q2, q3, . . . . . . correspond to energy eigenstates of a quantum system and
|q1| , |q2| , |q3| , . . . . . . correspond to energy eigenvalues of the quantum system with degeneracy.
Theorem [distribution of programs (i.e., “energy eigenstates”), Solovay]
#{ p | p ∈ Dom U & |p| ≤ n } = 2n−H(n)+O(1) for all n ∈ N.
(In statistical mechanics, this quantity represents “the number of states
below energy n”)
Here H(n) = H(the base-two representation of n).
If the energy eigenvalues of a quantum system distribute according to the
above distribution, then the following situation can realize:
If T is a computable real, the partial randomness of the values of
thermodynamic quantities at temperature T equals to T
in the quantum system.
31
Appendix
32
Proof of the fixed point theorem by energy E(T )
Theorem [general form of fixed point theorem] Let f : (0, 1) → R.
Suppose that f is a strictly increasing function and there is g : (0, 1)×N+ → R
which satisfies the following conditions:
(i) ∀ T ∈ (0, 1) limk→∞ g(T, k) = f (T ).
(ii) The mapping (Q ∩ (0, 1)) × N 3 (r, k) 7→ g(r, k) is computable.
(iii) ∀ T ∈ (0, 1) ∃ k0 ∈ N+ ∃ a ∈ N+ ∃ b, c, d ∈ N ∀ k ≥ k0
¯
¯a
¯
¯c
|p
|
|pk+1|
¯
¯ − k+1
¯
¯
−b
−
+d
T
T
≤ g(T, k + 1) − g(T, k) ≤ ¯pk+1¯ 2
.
¯pk+1¯ 2
(iv) ∀ T ∈ (0, 1) ∃ t ∈ (T, 1) ∃ k0 ∈ N+ ∃ c, d ∈ N ∀ k ≥ k0 ∀ x ∈ (T, t)
2−c(x − T ) ≤ g(x, k) − g(T, k) ≤ 2d(x − T ).
(v) ∀ t1, t2 ∈ (0, 1) with t1 < t2 ∃ k0 ∈ N+ ∀ k ≥ k0 ∀ x ∈ [t1, t2] g(x, k) ≤ f (x).
(vi) ∀ k ∈ N+ ∀ T ∈ (0, 1) limx→T +0 g(x, k) = g(T, k).
Then, for every T ∈ (0, 1), if f (T ) is computable, then T is Chaitin T -random
and T -compressible.
33
Proof of the fixed point theorem by energy E(T )
Theorem [fixed point theorem by energy E(T )]
[posted again]
For every T ∈ (0, 1), if E(T ) is computable, then T is Chaitin T -random
and T -compressible.
A portion of the proof to confirm the condition (iv) :
Using the mean value theorem and the lemma below, for every k ∈ N+ and
every T, x, t ∈ (0, 1) with T < x < t, there exists y ∈ (T, x) such that
Ek (x) − Ek (T ) = Ck (y)(x − T ).
On the other hand, for every T1, T2 ∈ (0, 1) with T1 < T2, there exist a ∈ N
and k0 ∈ N+ such that, every k ≥ k0 and every y ∈ [T1, T2],
0 < min Ck0 ([T1, T2]) ≤ Ck (y) < max C([T1, T2]).
Lemma [thermodynamic relations] T ∈ (0, 1).
(i) Fk0 (T ) = −Sk (T ), Ek0 (T ) = Ck (T ), and Sk0 (T ) = Ck (T )/T .
(ii) F 0(T ) = −S(T ), E 0(T ) = C(T ), and S 0(T ) = C(T )/T .
(iii) Sk (T ), Ck (T ) ≥ 0. Sk (T ), Ck (T ) > 0 for all sufficiently large k.
S(T ), C(T ) > 0.
34
Relation between the sufficient conditions of FPTs II
Theorem
There does not exist T ∈ (0, 1) such that all of Z(T ), E(T ),
and S(T ) are computable.
Proof) Use the statistical mechanical relation
S(T ) =
Theorem
E(T )
+ log2 Z(T ).
T
There does not exist T ∈ (0, 1) such that all of F (T ), E(T ),
and S(T ) are computable.
Proof) Use the thermodynamic relation
E(T ) − F (T )
.
S(T ) =
T
35
Some other property of the sufficient condition in FPTs
Using the fixed point theorem by Z(T ), some property of the computability
of Z(T ) is derived.
Let T ∈ (0, 1) and a ∈ (0, 1]. Assume that a is computable.
H((aT ) ¹ n)
H(T ¹ n)
= aT ⇒ lim
= aT .
n→∞
n→∞
n
n
by FPT
by H((aT ) ¹ n) = H(T ¹ n) + O(1)
Z(aT ) is computable
Theorem
⇒
lim
Sa ∩ Sb = ∅ for any distinct computable reals a, b ∈ (0, 1], where
Sa = { T ∈ (0, 1) | Z(aT ) is computable }.
Example
For every T ∈ (0, 1), if Z(T ) is computable, then for each integer
n ≥ 2, Z(T /n) is not computable. Namely,
for every T ∈ (0, 1), if the sum
sum
∑
(
2−|p|/T
)n
∑
2−|p|/T is computable, then its power
p∈Dom U
is not computable for every integer power n ≥ 2.
p∈Dom U
36